From here onwards we follow the steps of [1]}, [2]} and develop machinery needed for the proof of Theorem REF . We recall that we have isometrically embedded \(M\)  into the closed Riemannian manifold \((N,g)\) . The maximal subset \(M_p \subset T_pM\)  where the exponential map \(\exp _p\colon M_p \rightarrow M\)  of \((M,g)\)  is well defined was given in (REF ) as
\(M_p=\left\lbrace X\in T_pM: \: X=0 \hbox{ or } \Vert X\Vert _g\le {\tau _{\text{exit}}}\left(p,\frac{X}{\Vert X\Vert _{g}}\right)\right\rbrace .\) 
